First, start by hydrating the yeast by stirring it with the warm milk in the bowl of an electric mixer fitted with the paddle attachment. Let stand for 8 minutes.
Add the flour and salt, and mix together on low until you get a dry, shaggy mixture.
Add the eggs, one at a time, along with the sugar, beating between each addition.
Switch to the dough hook attachment and beat the dough until you get a smooth dough and the bowl is clean (scrape down the dry bits stuck to the sides as you go to be sure that all the ingredients get incorporated into the dough). This will take about 3 to 4 minutes.
Switch back to the paddle and dump in all the butter, and beat it into the dough. The dough will break down and become sticky and messy, but when it starts to form a more cohesive mass, switch back to the dough hook and beat on low (speed 2 on a KitchenAid mixer) for about 10 minutes or so, or until the dough is smooth and elastic, and the bowl is clean again.
Butter a bowl and transfer the dough to the bowl. Cover with saran wrap. Let rise on the counter until it has doubled (I left mine for almost 3 hours), then punch it down to deflate, cover, and leave in the fridge overnight.
The next morning, divide the dough into 8 pieces, and quickly roll them on the counter to make dough balls. Don’t work the dough too long or you risk melting the butter. Butter 8 slots of a muffin tin and place a dough ball in each. Place the muffin pan on a rimmed baking sheet and cover with a sheet of waxed paper. Put the sheet in the oven (turned off!) with just the light on. Let the dough balls rise until they have doubled (about 1.5 hours). When the dough has doubled, take the pan out of the oven and set it on the counter while you preheat the oven to 400ºF.
Meanwhile, mix the egg with 1 tsp water and brush lightly onto each bun (there will be leftover egg wash—you don’t need to use it all). When the oven is preheated, bake for 15–20 minutes, until the tops are a deep golden brown. Let cool 10 minutes before unmolding. Serve warm with homemade jam or marmalade, and more butter, if you dare.